Bottom line is that tinycdi is not an official release but a compilation of a certain driver at a certain time as a snapshot. Strictly speaking, MAMETESTERS only logs bugs/regressions that happened between official releases. In this case, MESS 0.134 and MESS 0.135. Sadly, CDI didn't work at all on 0.134 (setname CDI) and in 0.135 (setname CDIMONO1), it shows the same problem as it does now.
I was able to get the Zelda's Adventure (zeldaadv) to work with tinycdi with extracting the current CHD in circulation in softlist XML (version 5) and using MAME/MESS's tools from the era (0.134/0/135) CHDMAN to make a version specific (version 3) CHD - so that is your only recourse at this time.